Flag-accurate arrangements

verfasst von

Paul Mücksch, Gerhard Röhrle, Tan Nhat Tran

Abstract

In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let \(\mathscr A\) be a free arrangement of rank \(\ell\). Suppose that for every \(1\leq d \leq \ell\), the first \(d\) exponents of \(\mathscr A\) -- when listed in increasing order -- are realized as the exponents of a free restriction of \(\mathscr A\) to some intersection of reflecting hyperplanes of \(\mathscr A\) of dimension \(d\). Following [MR21], we call such an arrangement \(\mathscr A\) with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call \(\mathscr A\) flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

Ruhr-Universität Bochum
Kyushu University

Typ

Preprint

Publikationsdatum

01.02.2023

Publikationsstatus

Elektronisch veröffentlicht (E-Pub)